194 THE RATE OF GROWTH [ch. 



stalk the growth-rate falls away ; the florets are of descending age, 

 from flower tO' bud: their graded differences of age lead to an 

 exquisite gradation. of size and form; the time-interval between 

 one and another, or the "space-time relation" between them all, 

 gives a peculiar quality — ^we may call it phase-beauty — to the 

 whole. A clump of reeds or rushes shews this same phase-beauty, 

 and so do the waves on a cornfield or on the sea. A jet of water 

 is not much, but a fountain becomes a beautiful thing, and the 

 play of many fountains is an enchantment at Versailles. 



On the weight-length coefficient, or ponderal itidex 



So much for the visible changes of form which accompany 

 advancing age, and are brought about by a diversity of rates of 

 growth at successive points or in different directions. But it often 

 happens that an animal's change of form may be so gradual as to 

 pass unnoticed, and even careful measurement of such small changes 

 becomes difficult and uncertain. Sometimes one dimension is easily 

 determined, but others are hard to measure with the same accuracy. 

 The length of a fish is easily measured ; but the breadth and depth 

 of plaice or haddock are vaguer and more uncertain. We may then 

 make use of that ratio of weight to length which we spoke of in the 

 last chapter: viz. that W oc L^, or W ^ kL^, or W/L^ = k, where 

 k, the "ponderal index," is a constant to be determined for each 

 particular case*. 



We speak of this /; as a "constant," with a mean value specific 

 to each species of animal and dependent on the bodily proportions 

 or form of that animal; yet inasmuch as the animal is continually 

 apt to change its bodily proportions during life, k also is continually 

 subject to change, and is indeed a very delicate index of such 



* This relation, and how important it is, were clearly recognised by Herbert 

 Spencer in his Recent Discussions in Science, etc., 1871. The formula has been 



X y/w 



often, and often independently, employed: first perhaps in the form — y- x 100, 



by R. Livi, L'indice ponderale, o rapporto tra la statura e il peso, Atti Soc. Romamt 

 Antropologica, v, 1897. Values of k for man and many animals are given by 

 H. Przibram, in Form und Formel, 1922. On its use as an index to the condition 

 or habit of body of an individual, see von Rhode, in Abderhalden's Arbeitsmethoden, 

 IX, 4. The constant k might be called, more strictly, ki, leaving kf, and k,i for 

 the similar constants to be deri-ved from the breadth and depth of the fish. 



